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A132729 Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows. 3

%I

%S 1,1,1,1,1,1,1,3,3,1,1,5,9,5,1,1,7,17,17,7,1,1,9,27,37,27,9,1,1,11,39,

%T 67,67,39,11,1,1,13,53,109,137,109,53,13,1,1,15,69,165,249,249,165,69,

%U 15,1,1,17,87,237,417,501,417,237,87,17,1,1,19,107,327,657,921,921,657,327,107,19,1

%N Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows.

%H G. C. Greubel, <a href="/A132729/b132729.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n, k) = 2*A132044(n, k) - 1.

%F From _G. C. Greubel_, Feb 13 2021: (Start)

%F T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1.

%F Sum_{k=0..n} T(n, k) = 2^(n+1) - 3*n + 1 - 2*[n=0] = A132730(n). (End)

%e First few rows of the triangle are:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 3, 3, 1;

%e 1, 5, 9, 5, 1;

%e 1, 7, 17, 17, 7, 1;

%e 1, 9, 27, 37, 26, 9, 1;

%e 1, 11, 39, 67, 67, 39, 11, 1;

%t T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 3];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 13 2021 *)

%o (Sage)

%o def T(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) - 3

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 13 2021

%o (Magma)

%o T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 3 >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 13 2021

%Y Cf. A007318, A132044, A132730.

%K nonn,tabl

%O 0,8

%A _Gary W. Adamson_, Aug 26 2007

%E More terms added by _G. C. Greubel_, Feb 13 2021

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Last modified August 5 21:02 EDT 2021. Contains 346488 sequences. (Running on oeis4.)